Assessment of Two Surface Monte Carlo (TSMC) method to find stationary points of (H2O)15 and (H2O)20 clusters

The Two Surface Monte Carlo (TSMC) technique reduces computational cost by using a computationally cheap biasing potential, which guides the molecular system to explore the potential energy surface of interest. It was shown earlier that the Effective Fragment Potential (EFP) can be a good choice for this biasing potential (Bandyopadhyay, J Chem Phys 122:091102, 2005) when the potential energy surface of interest is quantum mechanical. This may help in expanding the applicability of TSMC, since finding a good biasing potential is a major challenge. In the present work, the viability of TSMC method in finding stationary points of large molecular system is investigated using EFP as the biasing potential and RHF theory as the potential of interest. TSMC is applied to find the stationary points of water clusters of size 15 and 20. A semi-automated method starting from random geometries, without using any chemical intuition, found several stationary points. The simulated annealing method was used to refine the structures obtained from TSMC. Among the several low-energy structures obtained for 15 water cluster, one minimum, about 1 kcal/mol higher than the global minimum, was found. However, for 20 water cluster, no structure very close to the global minimum was obtained. Several strategies, learned from the experience of the present work, are discussed for improving the TSMC method, including the acceptance between the two energy surfaces. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Gradient tabu search

This paper presents a modification of the tabu search called gradient tabu search (GTS). It uses analytical gradients for a fast minimization to the next local minimum and analytical diagonal elements of the Hessian to escape local minima. For an efficient blocking of already visited areas tabu regions and tabu directions are introduced into the tabu list (TL). Trials with various well-known test functions indicate that the GTS is a very promising approach to determine local and global minima of differentiable functions. Possible application areas could be optimization routines for force field parameters or conformational searches for large molecules.     

General methodology in two dimensions for classical simulation of reactive and nonreactive events on ab initio potential energy surfaces

A completely general two-dimensional (2D) methodology for the classical simulation of reactive and nonreactive events on ab initio potential energy surfaces is introduced and tested. The methodology requires the minimum amount of information given a priori—geometries and energies at these geometries. From a list of ab initio geometries and energies, simulations may be executed and a distribution of outcomes obtained. The method introduced attempts a local approach at simulating the dynamics of the system, rather than a global analytic fit to the potential energy surface. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1431–1444, 1998     

Global optimization of binary Lennard-Jones clusters using three perturbation operators

Global optimization of binary Lennard-Jones clusters is a challenging problem in computational chemistry. The difficulty lies in not only that there are enormous local minima on the potential energy surface but also that we must determine both the coordinate position and the atom type for each atom and thus have to deal with both continuous and combinatorial optimization. This paper presents a heuristic algorithm (denoted by 3OP) which makes extensive use of three perturbation operators. With these operators, the proposed 3OP algorithm can efficiently move from a poor local minimum to another better local minimum and detect the global minimum through a sequence of local minima with decreasing energy. The proposed 3OP algorithm has been evaluated on a set of 96 × 6 instances with up to 100 atoms. We have found most putative global minima listed in the Cambridge Cluster Database as well as discovering 12 new global minima missed in previous research.     

Shear-induced disappearances of energy minima and plastic deformation in polymer glasses

The potential energy landscape of a polymer glass is examined with regard to plastic deformation under shear strain. Shear strain is found to cause the disappearance of local potential energy minima, as determined by the decrease to zero of the curvature of the local minima. If the local energy minimum which the system is in disappears, the system becomes mechanically unstable and is forced to relax to an alternate energy minimum. This mechanical instability, which leads to a discontinuous change in system properties, is inherently irreversible—the new local minimum will not disappear, in general, when the strains are reversed. These disappearances of local energy minima and the associated irreversible relaxations lead to plastic deformation in polymer glasses.     

A gradient extremal walking algorithm

Gradient extremals define stream beds connecting stationary points on molecular potential energy surfaces. Using this concept we have developed an algorithm to determine transition states. We initiate walks at equilibrium geometries and follow the gradient extremals until a stationary point is reached. As an illustration we have investigated the mechanism for exchange of protons on carbon in methylenimine (H₂C=NH) using a multi-reference self-consistent-field wave function.     

A reliable prediction of the global phase stability for liquid–liquid equilibrium through the simulated annealing algorithm: Application to NRTL and UNIQUAC equations

The simulated annealing algorithm is introduced to search the global optimal solutions for the multipeak phenomena which generally exist in the phase stability problems with continuous variables. The Gibbs free energy criterion was modeled by the NRTL and UNIQUAC activity coefficient equations. When previous approaches fail, it is usually because they locate local minima due to the nonconvex and nonlinear natures of the models used to predict phase equilibrium. In this paper, the preliminary results show that the global minimum of the tangent plane distance function (TPDF) can be obtained by using the simulated annealing algorithm. The effects of the initial and the final values of the control parameter, the decrement of the control parameter and the length of the Markov chains are analyzed. The optimal `cooling schedule' was obtained according to the calculation results of the phase stability problems for one ternary mixture. The liquid–liquid equilibrium compositions were calculated by the Newton–Raphson method on the basis of the global minimum of TPDF. The results of four examples show that the simulated annealing algorithm can effectively solve the global phase stability problem.     

Stationary points on the H2CO potential energy surface: dependence on theoretical level

A total of 36 stationary points have been located on the H₂CO potential energy surface by means of gradient extremal following. These 36 points are believed to represent all the important stationary points on this surface. There is no indication that the structure of the surface becomes less complicated as the size of the basis set is enlarged at the Hartree-Fock level of theory, but many of the second- and third-order saddle points disappear when electron correlation is introduced. Of the ten first-order saddle points (transition structures) located, the majority have reaction paths entering the associated minima in a side-on approach, i.e. these cannot be located by uphill walking from the minimum. Received: 5 February 1998 / Accepted: 21 May 1998 / Published online: 29 July 1998     

Size‐guided multi‐seed heuristic method for geometry optimization of clusters: Application to benzene clusters

Since searching for the global minimum on the potential energy surface of a cluster is very difficult, many geometry optimization methods have been proposed, in which initial geometries are randomly generated and subsequently improved with different algorithms. In this study, a size‐guided multi‐seed heuristic method is developed and applied to benzene clusters. It produces initial configurations of the cluster with n molecules from the lowest‐energy configurations of the cluster with n − 1 molecules (seeds). The initial geometries are further optimized with the geometrical perturbations previously used for molecular clusters. These steps are repeated until the size n satisfies a predefined one. The method locates putative global minima of benzene clusters with up to 65 molecules. The performance of the method is discussed using the computational cost, rates to locate the global minima, and energies of initial geometries. © 2018 Wiley Periodicals, Inc.     

Comparing search strategies for finding global optima on energy landscapes

We provide some tests of the convex global underestimator (CGU) algorithm, which aims to find global minima on funnel-shaped energy landscapes. We use two different potential functions—the reduced Lennard–Jones cluster potential, and the modified Sun protein folding potential, to compare the CGU algorithm with the simplest versions of the traditional trajectory-based search methods, simulated annealing (SA), and Monte Carlo (MC). For both potentials, the CGU reaches energies lower on the landscapes than both SA and MC, even when SA and MC are given the same number of starting points as in a full CGU run or when all methods are given the same amount of computer time. The CGU consistently finds the global minima of the Lennard–Jones potential for all cases with up to at least n=30 degrees of freedom. Finding the global or near-global minimum in the CGU method requires polynomial time [scaling between O(n³) and O(n⁴)], on average. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1527–1532, 1999     

Powerful simulated-annealing algorithm locates global minimum of protein-folding potentials from multiple starting conformations

Protein-folding potentials, designed with the explicit goal that the global energy minimum correspond to crystallographically observed conformations of protein molecules, may offer great promise toward calculating native protein structures. Achieving this promise, however, depends on finding an effective means of dealing with the multiple-minimum problem inherent in such potentials. In this study, a protein-folding-potential test system has been developed that exhibits the properties of general protein-folding potentials yet has a unique well-defined global energy minimum corresponding to the crystallographically determined conformation of the test molecule. A simulated-annealing algorithm is developed that locates the global minimum of this potential in four of eight test runs from random starting conformations. Exploration of the energy-conformation surface of the potential indicates that it contains the numerous local minima typical of protein-folding potentials and that the global minimum is not easily located by conventional minimization procedures. When the annealing algorithm is applied to a previously developed actual folding potential to analyze the conformation of avian pancreatic polypeptide, a new conformer is located that is lower in energy than any conformer located in previous studies using a variety of minimization techniques.     

A strategy to find minimal energy nanocluster structures

An unbiased strategy to search for the global and local minimal energy structures of free standing nanoclusters is presented. Our objectives are twofold: to find a diverse set of low lying local minima, as well as the global minimum. To do so, we use massively the fast inertial relaxation engine algorithm as an efficient local minimizer. This procedure turns out to be quite efficient to reach the global minimum, and also most of the local minima. We test the method with the Lennard–Jones (LJ) potential, for which an abundant literature does exist, and obtain novel results, which include a new local minimum for LJ₁₃, 10 new local minima for LJ₁₄, and thousands of new local minima for . Insights on how to choose the initial configurations, analyzing the effectiveness of the method in reaching low‐energy structures, including the global minimum, are developed as a function of the number of atoms of the cluster. Also, a novel characterization of the potential energy surface, analyzing properties of the local minima basins, is provided. The procedure constitutes a promising tool to generate a diverse set of cluster conformations, both two‐ and three‐dimensional, that can be used as an input for refinement by means of ab initio methods. © 2013 Wiley Periodicals, Inc.     

Accurate MRCI and CC study of the most relevant stationary points and other topographical attributes for the ground-state C(2)H(2) potential energy surface

Extensive ab initio calculations have been performed to determine the energy, geometry, vibrational frequencies, and relative energetics of all stationary points of the C(2)H(2) ground-state potential-energy surface. The geometries of acetylene and vinylidene minima as well as all transition states are reported at the CASSCF, MRCI, and CCSD(T) levels with aug-cc-pVXZ basis sets. Other more advanced levels of CC theory have also been utilized where judged adequate, mostly for check purposes. Also reported are theoretical limiting values of the energetics of the reaction, deduced from series of computations using the USTE extrapolation method. The data here reported should be valuable for modeling a single-sheeted global potential energy surface for the title system.     

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.     

On the investigation of the low-energy conformational space of molecules

A new perspective on traditional energy minimization problems is provided by a connection between statistical thermodynamics and combinatorial optimization (finding the minimum of a function depending on many variables). The joint use of a new method for uncovering the global minimum of intramolecular potential energy functions, based on following the asymptotic behavior of a system of stochastic differential equations, and an iterative-improvement technique, whereby a search for relative minima is made by carrying out local quasi-Newton minimizations starting from many distinct points of the energy hypersurface, proved most effective for investigating the low-energy conformational space of molecules.     

Computer optimization of small atomic clusters

We have successfully identified stable configurations of both rare-gas and NaCl clusters with a new optimization procedure. An initial cluster configuration is prepared in a so-called shoot-and-stay process. Its total energy is then minimized with respect to the atomic coordinates. To prevent the system from being locked in local minima, the step size of each move is chosen as the width of the energy well at a higher level. As the system evolves, the global minimum is contained in the volume bounded by the decreasing value of step sizes. We have also carried out the optimization of NaCl clusters by the simulated annealing technique, for comparison. The results show that for such heterogeneous systems, the latter method cannot always find the global minimum, because of large energy gaps between different catchment regions in phase space.     

Alternative search strategy for minimal energy nanocluster structures: the case of rhodium, palladium, and silver

An alternative strategy to find the minimal energy structure of nanoclusters is presented and implemented. We use it to determine the structure of metallic clusters. It consists in an unbiased search, with a global minimum algorithm: conformational space annealing. First, we find the minima of a many-body phenomenological potential to create a data bank of putative minima. This procedure assures us the generation of a set of cluster configurations of large diversity. Next, the clusters in this data bank are relaxed by ab initio techniques to obtain their energies and geometrical structures. The scheme is successfully applied to magic number 13 atom clusters of rhodium, palladium, and silver. We obtained minimal energy cluster structures not previously reported, which are different from the phenomenological minima. Moreover, they are not always highly symmetric, thus casting some doubt on the customary biased search scheme, which consists in relaxing with density functional theory global minima chosen among high symmetry structures obtained by means of phenomenological potentials.